The Riemann solver for a strictly convex flux

The shock profile solves the equation and has the right trace. Assume $u_L>u_R$ and let $u$ be the profile (i) with speed $s=(f(u_R)-f(u_L))\mathrm{/}(u_R-u_L)$, so $s[u]=[f]$ with $[h]=h(u_R)-h(u_L)$: this is the Rankine--Hugoniot condition. By [F2] the weak residual of a piecewise constant profile with a single jump reduces to the interface integral $-\int \phi (t,st)([f]-s[u])dt$, which vanishes; hence $u$ is a distributional weak solution. Moreover $u(t,\cdot )$ differs from $u_0$ only on the interval between $0$ and $st$, whose length is $\mathrm{\mid }s\mathrm{\mid }t$, and $\mathrm{\mid }u-u_0\mathrm{\mid }\le \mathrm{\mid }{u}_L-u_R\mathrm{\mid }$ there, so ${\int }_K\mathrm{\mid }u(t,x)-u_0(x)\mathrm{\mid }dx\le \mathrm{\mid }{u}_L-u_R\mathrm{\mid }\mathrm{\mid }s\mathrm{\mid }t\to 0$ as $t\downarrow 0$: the strong local $L^1$ trace holds.

Regularized fans are weak solutions with vanishing entropy production. For $ϵ>0$ put $f_{ϵ}(r)=f(r)+\frac{ϵ}{2}{r}^2$, let ${\psi }_{ϵ}=(f_{ϵ}^{\mathrm{\prime }}{)}^{-1}$ on $[f_{ϵ}^{\mathrm{\prime }}(u_L),f_{ϵ}^{\mathrm{\prime }}(u_R)]$, and define $U_{ϵ}$ by the same three-branch formula with ${\xi }_L^{ϵ}=f_{ϵ}^{\mathrm{\prime }}(u_L)$, ${\xi }_R^{ϵ}=f_{ϵ}^{\mathrm{\prime }}(u_R)$, and $u^{ϵ}(t,x)=U_{ϵ}(x\mathrm{/}t)$. Each branch is $C^1$ by [F5], and on the open middle region the chain rule gives $u_t^{ϵ}+f_{ϵ}(u^{ϵ}{)}_x=\frac{1}{t}{\psi }_{ϵ}^{\mathrm{\prime }}(\xi )(f_{ϵ}^{\mathrm{\prime }}({\psi }_{ϵ}(\xi ))-\xi )=0$ since $f_{ϵ}^{\mathrm{\prime }}({\psi }_{ϵ}(\xi ))=\xi$; on the outer regions the profile is constant, so the residual vanishes pointwise there as well. At the two interfaces the traces of $u^{ϵ}$, hence of $f_{ϵ}(u^{ϵ})$, agree from both sides, so by [F2] no interface term arises and $u^{ϵ}$ is a distributional weak solution of $u_t+f_{ϵ}(u{)}_x=0$. The same computation applied to a convex $C^2$ pair $(\eta ,q_{ϵ})$ with $q_{ϵ}^{\mathrm{\prime }}={\eta }^{\mathrm{\prime }}{f}_{ϵ}^{\mathrm{\prime }}$ gives ${\mathrm{\partial }}_t\eta (u^{ϵ})+{\mathrm{\partial }}_x{q}_{ϵ}(u^{ϵ})={\eta }^{\mathrm{\prime }}(u^{ϵ})(u_t^{ϵ}+f_{ϵ}^{\mathrm{\prime }}(u^{ϵ})u_x^{ϵ})=0$ on each open branch and continuous traces $\eta (u^{ϵ}),q_{ϵ}(u^{ϵ})$ at the interfaces, so the entropy residual is identically $0$ for every convex $C^2$ pair.

The shock is entropic. With $u^{-}=u_L$, $u^{+}=u_R$ and speed $s$, the function $F(z)=f(z)-f(u_L)-s(z-u_L)$ satisfies $F(u_L)=F(u_R)=0$, and by strict convexity [F4] the graph of $f$ lies strictly below the chord through $(u_L,f(u_L))$, $(u_R,f(u_R))$ on $(u_R,u_L)$; that chord has slope $s$, so $F(z)<0$ for $z\in (u_R,u_L)$. Since $u_R-u_L<0$, $F(z)(u_R-u_L)\ge 0$ on $[u_R,u_L]$, and the chord criterion [F3] gives the entropy inequality for every convex $C^2$ pair.

The rarefaction profile and its inverse. Assume $u_L<u_R$ and set ${\xi }_L=f^{\mathrm{\prime }}(u_L)<{\xi }_R=f^{\mathrm{\prime }}(u_R)$; by [F4] the inverse $\psi =(f^{\mathrm{\prime }}{)}^{-1}\text{\hspace{0.17em}⁣}:[{\xi }_L,{\xi }_R]\to [u_L,u_R]$ is continuous and strictly increasing. Define $U(\xi )=u_L$ for $\xi \le {\xi }_L$, $U(\xi )=\psi (\xi )$ for ${\xi }_L\le \xi \le {\xi }_R$, and $U(\xi )=u_R$ for $\xi \ge {\xi }_R$, and set $u(t,x)=U(x\mathrm{/}t)$ for $t>0$. Then $u$ takes values in $[u_L,u_R]$, differs from $u_0$ only between $\min \{0,{\xi }_L\}t$ and $\max \{0,{\xi }_R\}t$, an interval of length $(\max \{0,{\xi }_R\}-\min \{0,{\xi }_L\})t$, and satisfies the strong local $L^1$ initial trace by the same estimate as in step 1.1.

Passage to the limiting fan. As $ϵ\downarrow 0$, $f_{ϵ}^{\mathrm{\prime }}\to f^{\mathrm{\prime }}$ uniformly on $[u_L,u_R]$ by [F5], so the clamped inverse profiles converge uniformly on $\mathbb{R}$. Indeed, if $D=\max \{\mathrm{\mid }{u}_L\mathrm{\mid },\mathrm{\mid }{u}_R\mathrm{\mid }\}$, then $\mathrm{\mid }{f}_{ϵ}^{\mathrm{\prime }}-f^{\mathrm{\prime }}\mathrm{\mid }\le ϵD$ on the state interval, so $U(\xi -ϵD)\le U_{ϵ}(\xi )\le U(\xi +ϵD)$. The extended continuous profile $U$ is uniformly continuous (it is constant outside a compact interval), yielding ${\sup }_{\xi }\mathrm{\mid }{U}_{ϵ}-U\mathrm{\mid }\to 0$; also $f_{ϵ}\to f$ and $q_{ϵ}\to q$ uniformly on the compact range $[u_L,u_R]$, where $q^{\mathrm{\prime }}={\eta }^{\mathrm{\prime }}{f}^{\mathrm{\prime }}$ is the flux of the same convex pair for $f$. Passing to the limit in the weak residual: $\mid {\int }_{{\mathrm{\Pi }}_T}(U_{ϵ}(x\mathrm{/}t)-U(x\mathrm{/}t)){\phi }_t\mid \le \sup \mathrm{\mid }{U}_{ϵ}-U\mathrm{\mid }\mathrm{\parallel }{\phi }_t{\mathrm{\parallel }}_1\to 0$ and $\mathrm{\parallel }{f}_{ϵ}(U_{ϵ})-f(U){\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{f}_{ϵ}-f{\mathrm{\parallel }}_{\mathrm{\infty },[u_L,u_R]}+\mathrm{L}\mathrm{i}\mathrm{p}(f)\sup \mathrm{\mid }{U}_{ϵ}-U\mathrm{\mid }\to 0$, so ${\int }_{{\mathrm{\Pi }}_T}(U{\phi }_t+f(U){\phi }_x)=0$ and $u$ is a weak solution. The entropy residual passes similarly, so for every convex $C^2$ pair $(\eta ,q)$ with $q^{\mathrm{\prime }}={\eta }^{\mathrm{\prime }}{f}^{\mathrm{\prime }}$ one has ${\int }_{{\mathrm{\Pi }}_T}(\eta (u){\phi }_t+q(u){\phi }_x)\ge 0$ for every nonnegative $\phi \in C_c^{\mathrm{\infty }}({\mathrm{\Pi }}_T)$.

Kruzhkov pairs by smoothing. Fix $k\in \mathbb{R}$ and let ${\eta }^{\delta }(r)=\sqrt{(r-k{)}^2+{\delta }^2}-\delta$, a smooth convex function with $0\le {\eta }^{\delta }(r)\le \mathrm{\mid }r-k\mathrm{\mid }$ and $\mathrm{\mid }{\eta }^{\delta }(r)-\mathrm{\mid }r-k\mathrm{\mid }\mathrm{\mid }\le \delta$, and let $q^{\delta }(s)={\int }_k^s({\eta }^{\delta }{)}^{\mathrm{\prime }}(r)f^{\mathrm{\prime }}(r)dr$, so that $(q^{\delta }{)}^{\mathrm{\prime }}=({\eta }^{\delta }{)}^{\mathrm{\prime }}{f}^{\mathrm{\prime }}$. Since $z\mapsto \mathrm{sgn}(z-k)$ is bounded and $f^{\mathrm{\prime }}$ is continuous on the compact range of the profiles, dominated convergence gives $q^{\delta }(s)\to {\int }_k^s\mathrm{sgn}(r-k)f^{\mathrm{\prime }}(r)dr=\mathrm{sgn}(s-k)(f(s)-f(k))=q_k(s)$ uniformly for $s$ in the profile range; and ${\eta }^{\delta }\to {\eta }_k$ uniformly there. Applying the entropy inequality of step 2.1 (shock case, via [F3]) or of step 2.3 (rarefaction case) to $({\eta }^{\delta },q^{\delta })$ and passing to the limit using uniform convergence and $\phi \in C_c^{\mathrm{\infty }}$ gives ${\int }_{{\mathrm{\Pi }}_T}({\eta }_k(u){\phi }_t+q_k(u){\phi }_x)\ge 0$ for every nonnegative $\phi$; hence both profiles satisfy all Kruzhkov entropy inequalities.

The constant case and uniqueness. If $u_L=u_R$, the constant $u\equiv u_L$ is a distributional weak solution with the exact trace, and its entropy production vanishes, so it is a Kruzhkov entropy solution; any bounded Kruzhkov entropy solution with the same constant datum equals it by order preservation applied in both directions. In the cases (i) and (ii), the profiles are bounded Kruzhkov entropy solutions with datum $u_0$ by steps 1.1, 2.1, 2.2–2.3 and 3.1, and any bounded Kruzhkov entropy solution with datum $u_0$ coincides with the profile almost everywhere on ${\mathrm{\Pi }}_T$ by Uniqueness, comparison and order preservation of entropy solutions applied in both directions. This proves existence, uniqueness, and the asserted profile in each case.